A Subclass of Analytic Functions Related tok-Uniformly Convex and Starlike Functions

Saqib Hussain,Muhammad Asad Zaighum,Akhter Rasheed,Maslina Darus

doi:10.1155/2017/9010964

Abstract

We investigate some subclasses ofk-uniformly convex andk-uniformly starlike functions in open unit disc, which is generalization of class of convex and starlike functions. Some coefficient inequalities, a distortion theorem, the radii of close-to-convexity, and starlikeness and convexity for these classes of functions are studied. The behavior of these classes under a certain modified convolution operator is also discussed.

Highlights

Let A be the class of all analytic functions f in open unit disc Δ = {z : |z| < 1}, normalized by f(0) = 0 and f󸀠(0) = 1
A function f ∈ S is uniformly convex if f(z) maps every circular arc γ contained in Δ with center ζ ∈ Δ onto a convex arc
The function f ∈ S is uniformly starlike if f(z) maps every circular arc γ contained in Δ with center ζ ∈ Δ onto a starlike arc with respect to f(ζ)

Summary

Introduction

In 1991, Goodman [1, 2] introduced the classes UCV and UST of uniformly convex and uniformly starlike functions, respectively. [13] El-Ashwah et al introduced two important subclass k − UCV(α, β) and k − UST(α, β) of kuniformly convex starlike functions as f ∈ k − UCV (α, β) ⇐⇒ Let Vη be the class of functions f ∈ S given in (1) for which arg(an) = π + (n − 1)η, n ≥ 2.

Results

Conclusion